Recalling the thermal field dynamics (TFD) introduced by Takahashi and
Umezawa [1]}, [2]}, [3]}, its elemental spirit is to convert the calculations
of ensemble averages for a mixed state \(\rho \) , \(\left\langle A\right\rangle =\mathtt {tr}\left( A\rho \right) /\mathtt {tr}\left( \rho \right) ,\)  to
equivalent expectation values with a pure state \(\left|0(\beta )\right\rangle \) , i.e.,
\(\left\langle A\right\rangle =\left\langle 0(\beta )\right|A\left|0(\beta )\right\rangle , \) 
